Author |
Topic: 3 JEE matrix problems (Read 1213 times) |
|
towr
wu::riddles Moderator
Uberpuzzler
Some people are average, some are just mean.
Gender: 
Posts: 13730
|
On behalf of navdeep1771 I'm posting the follow problems from the 2010 Joint Entrance Exam
|
| « Last Edit: Jul 14th, 2018, 1:09pm by towr » |
 IP Logged |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
navdeep1771
Newbie
Let your thoughts go beyond your imagination
Gender:
Posts: 28
|
 |
Re: 3 JEE matrix problems
« Reply #1 on: Jul 15th, 2018, 4:08am » |
 Quote  Modify
|
I was unable to post this image file. As it requires some minimum number of posts.
So towr posted this from my side.
Thanks towr 
Well these 3 problems are new to everyone (including towr).
|
|
IP Logged |
|
|
|
towr
wu::riddles Moderator
Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13730
|
|
Re: 3 JEE matrix problems
« Reply #2 on: Jul 15th, 2018, 7:18am » |
Quote Modify
|
I hadn't heard of skew-symmetric matrices before. But from the definition I'd say it's either a red herring or we're missing a statement that we're operating modulo p.
On the other hand, it doesn't really matter. For odd p, a has to be 0 in a skew-symmetric matrix, because only then a=-a, and b = -c, so det(Tp)=b^2, which isn't divisible by p.
Now if we try p=5, the only answer that fits turns out to be D: 2p-1
|
|
IP Logged |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
navdeep1771
Newbie
Let your thoughts go beyond your imagination
Gender:
Posts: 28
|
|
Re: 3 JEE matrix problems
« Reply #3 on: Jul 18th, 2018, 12:30am » |
Quote Modify
|
That's Correct.
Well mathematical proof goes like this:-
We must have a^2 − b^2 = kp
⇒ (a + b) (a − b) = kp
⇒ either a − b = 0 or a + b is a multiple of p
when a = b; number of matrices is p
and when a + b = multiple of p ⇒ a, b has p − 1
∴ Total number of matrices = p + p − 1
= 2p − 1.
But what about '43' and '44'?
|
|
IP Logged |
|
|
|
towr
wu::riddles Moderator
Uberpuzzler
Some people are average, some are just mean.
Gender:
Posts: 13730
|
|
Re: 3 JEE matrix problems
« Reply #4 on: Jul 19th, 2018, 10:55pm » |
Quote Modify
|
Since a < p and p is odd, then 2a is only divisible by p when a = 0
If a= 0 then det (A) is a multiple p only if b=0 or c=0, which are 2p - 1 cases
This seems irrelevant, because it's exactly what's not asked in either question 43 or 44
However, together with 43 and 44 it covers all possible A, so answer(43) + answer(44) + (2p - 1) must be p^3
Which means the answers to 43 and 44 should be C and D
|
| « Last Edit: Jul 19th, 2018, 10:57pm by towr » |
IP Logged |
Wikipedia, Google, Mathworld, Integer sequence DB
|
|
|
navdeep1771
Newbie
Let your thoughts go beyond your imagination
Gender:
Posts: 28
|
|
Re: 3 JEE matrix problems
« Reply #5 on: Jul 20th, 2018, 8:15am » |
Quote Modify
|
@towr
You are insane.
|
|
IP Logged |
|
|
|
|
RIDDLES SITE
WRITE MATH!
Home 
Help 
Search 
Members 
Login 
Register
Reply
Notify of replies
Send Topic
Print